Two identical springs (neglect their masses) are used to “play catch” with a small block of mass 200 g (see the figure).

Spring A is attached to the floor and compressed 10.0 cm with the mass on the end of it (loosely).

Spring A is released from rest and the mass is accelerated upward. It impacts the spring attached to the ceiling, compresses it 2.00 cm, and stops after traveling a distance of 30.0 cm from the relaxed position of spring A to the relaxed position of spring B as shown.

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``````````````````````_}-- 2cm
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``````30cm--[
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`````10cm--{[ ________
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(THANKS FOR HELPING)

To find the spring constant of each spring, we can use Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring.

Let's denote the spring constant of spring A as k1 and the spring constant of spring B as k2.

First, we can determine the force exerted by spring A when it is compressed by 10.0 cm. The force exerted by a spring can be calculated using the equation:

F = k1 * x1

where F is the force, k1 is the spring constant, and x1 is the displacement of the spring (in this case, 10.0 cm = 0.1 m).

We know that this force is used to accelerate the block of mass 200 g (0.2 kg) upward, so we can use Newton's second law of motion to calculate the acceleration of the block:

F = m * a

where F is the force, m is the mass, and a is the acceleration.

Substituting the values, we get:

k1 * x1 = m * a

k1 * 0.1 = 0.2 * a

Next, we need to find the force exerted by spring B when it is compressed by 2.00 cm. Similarly, we can use Hooke's Law:

F = k2 * x2

where F is the force, k2 is the spring constant, and x2 is the displacement of the spring (in this case, 2.00 cm = 0.02 m).

This force will bring the block to a stop after traveling a distance of 30.0 cm from the relaxed position of spring A to the relaxed position of spring B. We can use the work-energy principle to derive this equation:

work done by spring B = change in kinetic energy

0.5 * m * v^2 = work done by spring B

where v is the velocity of the block.

The work done by a spring can be calculated using the equation:

work done by spring = 0.5 * k2 * x2^2

Substituting the values, we get:

0.5 * m * v^2 = 0.5 * k2 * x2^2

Finally, we need to find the maximum compression of spring A. We can use the conservation of mechanical energy:

Initial potential energy of spring A = Final potential energy of spring A + Final potential energy of spring B

0.5 * k1 * x1^2 = 0.5 * k1 * 0^2 + 0.5 * k2 * x2^2

By solving these equations simultaneously, we can find the values of k1 and k2, which would give us the spring constant of each spring.